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Please use this identifier to cite or link to this item: http://hdl.handle.net/1842/5030

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Powell2011.pdfPhD thesis1.46 MBAdobe PDFView/Open
Title: Second order algebraic knot concordance group
Authors: Powell, Mark Andrew
Supervisor(s): Butler, Leo
Ranicki, Andrew
Issue Date: 28-Jun-2011
Publisher: The University of Edinburgh
Abstract: Let Knots be the abelian monoid of isotopy classes of knots S1 ⊂ S3 under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-Orr-Teichner [COT03] defined a filtration of C: C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . .The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5) is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance class of a knot to a pointed set (COT (C/1.5),U). We define an abelian monoid of chain complexes P, with a monoid homomorphism Knots → P. We then define an algebraic concordance equivalence relation on P and therefore a group AC2 := P/ ~, our second order algebraic knot concordance group. The results of this thesis can be summarised in the following diagram: . That is, we define a group homomorphism C → AC2 which factors through C/F(1.5). We can extract the two stage Cochran-Orr-Teichner obstruction theory from AC2: the dotted arrows are morphisms of pointed sets. Our second order algebraic knot concordance group AC2 is a single stage obstruction group.
Sponsor(s): Engineering and Physical Sciences Research Council (EPSRC)
Keywords: abelian monoid
knots
algebraic concordance group
single stage obstruction group
geometric topology
URI: http://hdl.handle.net/1842/5030
Appears in Collections:Mathematics thesis and dissertation collection

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